Matching with time-dependent treatments: A review and look forward

Abstract

Observational studies of treatment effects attempt to mimic a randomized experiment by balancing the covariate distribution in treated and control groups, thus removing biases related to measured confounders. Methods such as weighting, matching, and stratification, with or without a propensity score, are common in cross-sectional data. When treatments are initiated over longitudinal follow-up, a target pragmatic trial can be emulated using appropriate matching methods. The ideal experiment of interest is simple; patients would be enrolled sequentially, randomized to one or more treatments and followed subsequently. This tutorial defines a class of longitudinal matching methods that emulate this experiment and provides a review of existing variations, with guidance regarding study design, execution, and analysis. These principles are illustrated in application to the study of statins on cardiovascular outcomes in the Framingham Offspring cohort. We identify avenues for future research and highlight the relevance of this methodology to high-quality comparative effectiveness studies in the era of big data.

Keywords: longitudinal matching; new-user design; real-world evidence; time-dependent confounding; time-varying treatment.

Figure 1

Schema of enrollment across j=1,…,n _S longitudinal pseudo‐experiments. Time scale, s, represents time since first eligibility for treatment initiation. {j} represents the set of eligible patients at time s _j satisfying R _i(s _j)=1, ${ℰ}_i(s_j)=1$, and S _i≥s _j for i=1,…,n patients. Any patients with S _i=s _j are treated and the remainder with S _i>s _j are controls. Covariate history ${\mathit{Z}}_i^{*}(s_j)$ is available to the jth experiment. A, Generic scheme; variable follow‐up. B, Specific example; homogenous follow‐up

Figure 2

Follow‐up time scales and censoring dates. Time scale, s, represents time since first eligibility for enrollment. Censoring is purely administrative and occurs at a common time point, C _i=τ _s, for all patients. We are interested in studying outcomes after treatment initiation for a minimum of τ _t years. Treatment initiation times must be limited to s∈(0,τ). As before, {j} represents the set of eligible patients at time s _j satisfying R _i(s _j)=1, ${ℰ}_i(s_j)=1$, and S _i≥s _j for i=1,…,n patients. Any patients with S _i=s _j are treated and the remainder with S _i>s _j are controls in the jth experiment. A, Extended follow‐up. B, Restricted follow‐up

Figure 3

Illustration of two hypothetical patients across three time scales (discrete for simplicity). Available data range across calendar time 2006‐2012. The relevant time scale for matching (s) is time since diagnosis. Follow‐up for outcomes (time scale t) begins after treatment (Tx) initiation. Censoring time with respect to s is C _i(s) and with respect to t is C _i(t)

Figure 4

Balance check at examination 7. Absolute standardized mean differences are the difference in covariate means between two groups (treated vs untreated) divided by the standard deviation of the same covariate. Unadjusted: patients who initiate statins are compared to eligible controls at examination 7. Matched: after matching on a time‐dependent propensity score, patients who initiate statins and their matched controls at examination 7 are compared

Figure 5

Hazar d ratios for the treatment effect of statins on cardiovascular outcomes over 15 years follow‐up

Figure 6

Subgroup analysis of the ITT effect of statins on cardiovascular outcomes over 15 years follow‐up. Sequential stratification (dashed line) and Sequential Cox model (solid line)